Couldn't find any specifics regarding a candidate having a police record as it applies to the application process at USCGA. USNA has a question on their app about having been charged or convicted of any felony or misdemeanor and a description of the charges and or convictions. After the candidate receives an appointment or is admitted, then the following takes place as a further vetting of the candidate:

Police Record Checks
Each candidate receiving an offer of appointment is required to complete a routine
Police Record Check with his or her local law enforcement agencies. The Naval
Academy will send you forms that you must take to these offices. These background
checks are a precursor to the National Agency Check (NAC) that will be initiated
during Plebe Summer. The Police Record Check and the NAC are used to determine
your credibility and suitability for service and to grant a security clearance for access
to classified material.

I have not dealt specifically with service academy background checks, but have extensive experience with federal employee checks....so I assume there are similarities.

The key words are SUITABILITY and CREDIBILITY. You cannot lie about your background. If you are caught lying, (credibility) you will have a difficult time passing the check. On the other hand, only report on your form what is true. What I mean by that is that you need to find out exactly what, if anything, is on your record, and report it as accurately as possible. But don't leave out something that they will find later...it looks like you are trying to hide something (credibility). If you feel that an explanation is in order, you might be able to append one to your application.

If you perceive you might have any potential problems, I suggest that you contact a lawyer and find out exactly what you are dealing with, and how best to truthfully and honestly disclose it. Do this BEFORE you apply so that you can plan accordingly. It is far better to be proactive than reactive.

If selected for an appointment to CGA, you'll have to fill out a fairly detailed "Homeland Security" security clearance form. In fact some of your courses at campus require a security clearance to attend.
It all depends what you mean by police record.... robbery, hit and run versus speeding tickets.

Carrying a concealed weapon...that's a no-no. Make sure you are up front and honest about it, because if you are not, it WILL come back, especially at some point if you need SECRET or TOP SECRET clearance.

I had a classmate in swab summer that had some legal problems before he entered the Academy. He didn't tell admissions. When it came up (within 7 weeks of R-day), he was quickly disenrolled.

You're not only entering a branch of the US Armed Forces, but also a federal law enforcement agency....police records matter.